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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Multiplicative bijections of semigroups of interval-valued continuous functions


Author: Jesús Araujo
Journal: Proc. Amer. Math. Soc. 137 (2009), 171-178
MSC (2000): Primary 46J10; Secondary 46E05, 54D35.
Published electronically: July 1, 2008
MathSciNet review: 2439438
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Abstract: We characterize all compact and Hausdorff spaces $ X$ which satisfy the condition that for every multiplicative bijection $ \varphi$ on $ C(X, I)$, there exist a homeomorphism $ \mu : X \longrightarrow X$ and a continuous map $ p: X \longrightarrow (0, +\infty)$ such that

$\displaystyle \varphi (f) (x) = f(\mu (x))^{p(x)}$

for every $ f \in C(X,I)$ and $ x \in X$. This allows us to disprove a conjecture of Marovt (Proc. Amer. Math. Soc. 134 (2006), 1065-1075). Some related results on other semigroups of functions are also given.


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Additional Information

Jesús Araujo
Affiliation: Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Facultad de Ciencias, Avda. de los Castros, s. n., E-39071 Santander, Spain
Email: araujoj@unican.es

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09448-3
PII: S 0002-9939(08)09448-3
Received by editor(s): October 26, 2007
Received by editor(s) in revised form: December 10, 2007
Published electronically: July 1, 2008
Additional Notes: This research was partially supported by the Spanish Ministry of Science and Education (Grant number MTM2006-14786).
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.